In plasma processing applications, such as the manufacture of semiconductors or flat panel displays, RF power generators apply a voltage to a load in a plasma chamber and may operate over a wide range of frequencies. The impedance of a processing plasma can vary with the frequency of this applied voltage, chamber pressure, gas composition, and the target or substrate material. And the impedance of the plasma load affects the efficiency at which power is applied from the generator to the load. Consequently, an estimate of the plasma load impedance is a parameter that is often desirable for users to have available.
Obtaining a good estimate of load impedance, however, is often difficult. For example, accurate measurements of forward and reflected power, more precisely the incident and reflected signals who's magnitudes squared are proportional to forward and reflected power, as well as the phase relationship between the forward and reflected signals may be utilized to obtain an estimate of load impedance, but when the measurement system (the measurement system to obtain forward and reflected signals) is not synchronized with a reference oscillator (e.g., an oscillator of the RF generator), each sampled measurement of forward and reflected power may potentially have a random phase with respect to the reference oscillator. As a consequence, it is very difficult, if not impossible, to average either the forward voltage or the reflected voltage measurements (e.g., to remove noise and unwanted modulation).
To illustrate the problem, consider a simple measurement system such as shown in FIG. 1A containing a sensor such as a directional coupler or voltage/current (VI) sensor. It can be shown that almost any linear four port network can be used as a sensor for determining forward and reflected power as well as load impedance. Such a sensor is perfectly correctable in the sense that there are four complex numbers such that
            [                                                  V              forward                                                                          V              reflected                                          ]        =                  [                                                            k                11                                                                    k                12                                                                                        k                21                                                                    k                22                                                    ]            ⁡              [                                                            V                3                                                                                        V                4                                                    ]                  as    ⁢                  ⁢    long    ⁢                  ⁢    as              S      13        ,                  S        24            ≠              S        14              ,          S      23      
where Sij, i, jε{1,2} are the scattering parameters (S parameters) of the sensor with the ports numbered as in FIG. 1A. In the above matrix equation Vforward and Vreflected are the corrected forward and reflected signals with respect to a 50′Ω reference impedance such that the forward power is equal to |Vforward|2, the reflected power is equal to |Vreflected|2 and the load reflection coefficient, ρload, is equal to Vreflected/Vforward. The load reflection coefficient in turn is related to the load impedance by
      Z    load    =      50    ⁢                            1          +                      ρ            load                                    1          -                      ρ            load                              .      
The entries in the matrix, k11, k12, k21 and k22 can be calculated from the scattering parameters of the sensor and the impedance presented to the sensor at the sense ports. Any one of the entries, typically k11 can be made real by multiplying through with a suitable complex number. Normally the entries are determined by calibrating the entire system. Such calibration can be performed by measuring the response of the sensor system to at least three impedances and using a power standard to scale the matrix correctly. Note that all numbers involved are complex numbers representing the magnitude and phase of the signals in a convenient mathematical form. Thus, with t representing time and choosing an arbitrary instant in time to correspond to t=0, the signal V3 in the matrix equation is related to the time domain signal at port 3 of the sensor, ν3, by the equation:ν3(t)=V3ejω0t+V*3e−jω0t 
where ω0 is the frequency of the source in rad/s and x* represents the complex conjugate of x. The same holds true for V4, Vforward and Vreflected. As shown in FIG. 1A estimates of V3 and V4 can be obtained by simply taking samples of ν3 and ν4 90 degrees apart with 90 degrees corresponding to a delay of one fourth of the period of the source. A simple measurement system such as shown in FIG. 1A assumes a pure sinusoidal source. In general, a more complex system incorporating filters and more sophisticated and less noisy estimates of the phasors V3 and V4 are used. As illustrated in FIGS. 1B and 1C, unless the samples to estimate V3 and V4 are perfectly synchronized with the frequency of the source, samples of V3 and V4 taken at different times are rotated. In FIG. 1B, a(1)+jb(1) illustrates a sample of V3 taken at time t1, c(1)+jd(1) illustrates a sample of V4 taken at time t1, and in FIG. 1C, a(2)+jb(2) illustrates a sample of V3 taken at time t2, and c(2)+jd(2) illustrates a sample of V4 taken at time t2.
As shown in this illustration, the magnitudes of the samples of V3 and V4 and well as the phase relationship between V3 and V4 are determined by the power delivered to the load and the load impedance and do not change under steady state excitation, but the samples are rotated with respect to each other except in the special case where the sampling times are perfectly synchronized with the frequency of the source and taken exactly one or multiples of one cycle apart. The same is true for samples of the corrected forward and reflected signals Vforward and Vreflected. If averaging is used it can be applied to either the uncorrected signals V3 and V4 or the corrected signals Vforward and Vreflected. The choice between averaging depends on the available computational resources. It is often possible to calculate the corrected signals and carry out averaging on the corrected signals, but if computational resources are really limited it may be more advantageous to average the uncorrected signals V3 and V4 and perform slightly more computations at a much reduced rate to obtain the corrected signals from the averaged uncorrected signals.
One approach to deal with the problem of random phase in power measurements includes calculating, for each sampled pair of forward and reflected signals, a reflection coefficient, which is equal to the ratio of the reflected signal to the forward signal. Then the set of calculated reflection coefficients is averaged to obtain an average reflection coefficient value. For example, in one millisecond, a thousand measurements of forward and reflected signals may be taken, and as a consequence, a thousand division operations (e.g., uncorrected reflected signal V4(k) divided by uncorrected forward signal V3(k), kε{1, 2, . . . }) are carried out in each millisecond to obtain a set of reflection coefficient values that are then averaged to obtain an average reflection coefficient. Problematically, each time a reflection coefficient is calculated in this manner, system resources are utilized; thus this approach to obtaining an average reflection coefficient is computationally intensive and is prone to excessive utilization of system resources.
As a consequence, known techniques are often too inefficient to provide desirable information about the electrical characteristics of plasma loads. Accordingly, a system and method are needed to address the shortfalls of present technology and to provide other new and innovative features.